Orthogonal Projection is Mapping
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Theorem
Let $H$ be a Hilbert space.
Let $K$ be a closed linear subspace of $H$.
Let $P_K: H \to H$ be the orthogonal projection on $K$.
Then $P_K$ is a mapping.
Proof
For $P_K$ to be a mapping we need to show that:
- $\forall h \in H: \map{P_K} h$ exists and is unique
By definition of $\map{P_K} h$, this amounts to:
- There is a unique $k \in K$ such that $\norm{ h - k } = \map d {h, K}$
This is precisely the statement of Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality